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Amplitude and Vertical Dilation of Sine and Cosine Functions
In this activity, you will discover the relationship between the vertical dilation A and the amplitude of: [pic] and [pic]
Applet: Amplitude_Vertical_Dilation
Terminology: Vertical dilation stretches or compresses the function along the y-axis.
[pic]
Before you begin, check that: A=1 and that the checkbox “Allow Fractions for A” is unchecked.
Questions and Answers:
1. Write down the parent functions of [pic] and[pic]. What is the value of A and what is the amplitude of these parent functions?
Answer: Parent functions are y = sin(x) and to y = cos(x), respectively. In both parent functions, the value of A is 1 and the amplitude=1.
2. Move slider A in any direction (A≠0). What type of transformation does this represent?
Answer: Moving A (A≠0) in any direction gives a vertical dilation or (since |A|>1), the transformation is a vertical stretch.
3. What happens to y = Asin(x) and to y = Acos(x) when A increases in the positive direction?
Answer: When A increases in the positive direction and since A>1, the functions stretch vertically more and more.
4. Set A = 2. Write the equations for both the sine and cosine functions below.
Answer: The equations are y = 2sin(x) and y = 2cos(x).
5. What is the amplitude of both sine and cosine under this vertical dilation?
Answer: The amplitudes of y = 2sin(x) and y = 2cos(x) is amplitude=2.
6. Set A = 4. Write the equations for both the sine and cosine functions below.
Answer: The equations are y = 4sin(x) and y = 4cos(x).
7. What is the amplitude of both sine and cosine under this vertical dilation?
Answer: The amplitudes of y = 4sin(x) and y = 4cos(x) is amplitude=4.
There seems to be a relationship between the amplitude of a sinusoidal function and A.
Let's see if you can see the pattern.
Table 1: Positive Whole Numbers for A.
|Dilation Factor |A = 1 |A = 2 |A = 4 |
|Amplitude |amplitude = 1 |amplitude = 2 |amplitude = 4 |
8. Write an equation for the amplitude in terms of A when A>0.
Answer: The amplitude of the functions y = Asin(x) and to y = Acos(x) is amplitude=A.
Note: Because we are not looking at A1, the functions y = Asin(x) and y = Acos(x) are stretched compressed and the amplitude A of these functions is amplitude > 1 amplitude < 1.
3. When the amplitude < 1 of the functions y = Asin(x) and y = Acos(x), this means that |A|>1 0 ................
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